The displacement of a viscous fluid by an air bubble in the narrow gap between two parallel plates can readily drive complex interfacial pattern formation known as viscous fingering. We focus on a modified system suggested recently by Zheng et al. [Phys. Rev. Lett. 115, 174501 (2015)], in which the onset of the fingering instability is delayed by introducing a time-dependent (power-law) plate separation. We perform a complete linear stability analysis of a depth-averaged theoretical model to show that the plate separation delays the onset of nonaxisymmetric instabilities, in qualitative agreement with the predictions obtained from a simplified analysis by Zheng et al. We then employ direct numerical simulations to show that in the parameter regime where the axisymmetrically expanding air bubble is unstable to nonaxisymmetric perturbations, the interface can evolve in a self-similar fashion such that the interface shape at a given time is simply a rescaled version of the shape at an earlier time. These novel, self-similar solutions are linearly stable but they only develop if the initially circular interface is subjected to unimodal perturbations. Conversely, the application of nonunimodal perturbations (e.g., via the superposition of multiple linearly unstable modes) leads to the development of complex, constantly evolving finger patterns similar to those that are typically observed in constant-width Hele-Shaw cells.

In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control theory. In this paper, we consider the problem of design and implementation of a controller for fully nonlinear, high-dimensional, dynamical systems with the goal of steering trajectories to an unstable fixed point of the governing equations. Our control strategy is based on our previous work [A. Blanchard, S. Mowlavi, and T. P. Sapsis, Nonlinear Dynam. 95, 2745 (2019)] and takes advantage of the unique properties of the optimally time-dependent (OTD) modes, a set of global, time-evolving, orthonormal modes that track directions in phase space associated with transient growth and persistent instabilities. We show that the OTD control strategy introduced previously is robust with respect to perturbation amplitude even in cases in which the trajectory initially evolves on an attractor that lies far away from the target fixed point. In recognition of the fact that actuation capabilities are generally limited in practice, we also formulate a localized control strategy in which the OTD modes are computed in a spatially localized subdomain of the physical domain of interest. We suggest a strategy for selecting the optimal control domain based on a quantitative criterion derived from the OTD modes. We show that even when the range of the controller is reduced, OTD control is able to steer trajectories toward the target fixed point.

Ongoing development of fuel storage and delivery systems for space probes, interplanetary vehicles, satellites, and orbital platforms continues to drive interest in propellant management systems that utilize surface tension to retain, channel, and control flow in microgravity environments. Although it has been known for decades that capillary flows offer an ideal method of fuel management, there has been little research devoted to the general stability properties of such flows. In this work we demonstrate theoretically why capillary flows which channel wetting liquids in slender open triangular channels tend to be very stable against disturbances. By utilizing the gradient flow form of the governing fluid interface equation, we first prove that stationary interfaces in the presence of steady flow are asymptotically nonlinearly and exponentially stable in the Lyapunov sense. We then demonstrate that fluid interfaces exhibiting self-similar Washburn dynamics are transiently and asymptotically linearly stable to small perturbations. This second finding relies on a generalized nonmodal stability analysis due to the non-normality of the governing disturbance operator. Taken together, these findings reveal the robust nature of transient and steady capillary flows in open grooved channels and likely explain the prevalent use of capillary flow management systems in many emerging technologies ranging from CubeSats to point-of-care microfluidic diagnostic systems.

In experiments and numerical simulations we measure angles between the symmetry axes of small spheroids advected in turbulence (passive directors). Since turbulent strains tend to align nearby spheroids, one might think that their relative angles are quite small. We show that this intuition fails in general because angles between the symmetry axes of nearby particles are anomalously large. We identify two mechanisms that cause this phenomenon. First, the dynamics evolves to a fractal attractor despite the fact that the fluid velocity is spatially smooth at small scales. Second, this fractal forms steps akin to scar lines observed in the director patterns for random or chaotic two-dimensional maps.

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed to be small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-Hückel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $\chi \ll 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender-body theory. In the opposite limit $\chi \gg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q={\epsilon }_1/{\epsilon }_2$, where ${\epsilon }_1,{\epsilon }_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

Using direct numerical simulations, we examine the effects of the Taylor Reynolds number, $R_{\lambda }\equiv u^{\prime }\lambda /\nu$ (where $u^{\prime }$, $\lambda$, and $\nu$ denote the fluid root-mean-squared velocity, the Taylor microscale and the fluid kinematic viscosity, respectively), and Froude number, $\text{Fr}\equiv a_{\eta }/g$ (where $a_{\eta }$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), on the motion of small, spherical, settling, bidisperse inertial particles (characterized by their Stokes number $\text{St}\equiv {\tau }_p/{\tau }_{\eta }$, which is the ratio of the particle response time to the Kolmogorov timescale) in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing Fr leads to an enhancement of the accelerations but a suppression of their intermittency. For Stokes numbers $\text{St}>1$, the effect of $R_{\lambda }$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales. The results for the probability density function (PDF) of the particle relative velocities show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as $R_{\lambda }$ is increased. This occurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the turbulence contribution to the particle motion is of the same order as that from gravitational settling. Increasing $R_{\lambda }$ enhances the non-Gaussianity of the relative velocity PDFs, while reducing Fr has the opposite effect, and for fast settling particles, the PDFs become approximately Gaussian. Finally, we observe that low-order statistics such as the radial distribution function and the particle collision kernel are strongly affected by Fr and St, and especially by the degree of bidispersity of the particles. However, we also find that these low-order statistics are very weakly affected by $R_{\lambda }$ when $\text{St}\le O\left(1\right)$, irrespective of the degree of bidispersity. Therefore, although the mechanisms controlling the collision rates of monodisperse and bidisperse particles are different, they share the property of a weak sensitivity to $R_{\lambda }$ when $\text{St}\le O\left(1\right)$.

A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by and regulates a potential enstrophy field, $\varepsilon$. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analyzed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of metastable quasiperiodic configurations, lasting for hundreds of time units, yet interspersed with abrupt ($\mathrm{\Delta }t\ll 1$) mergers of adjacent steps in the staircase. The mergers occur at the staircase lattice defects where the pattern has not completely relaxed to a strictly periodic solution that can be obtained analytically. Other types of stationary solutions are solitons and kinks in the PV gradient and $\varepsilon$-profiles. The waiting time between mergers increases strongly as the number of steps in the staircase decreases. This is because of an exponential decrease in interstep coupling strength with growing spacing. The long-time staircase dynamics is shown numerically be determined by local interaction with adjacent steps. Mergers reveal themselves through the explosive growth of the turbulent PV flux, which, however, abruptly drops to its global constant value once the merger is completed.

We investigated the generation and propagation of ultrasonic pressure waves produced by focused x-ray free-electron laser pulses in 14 to $30\mu \mathrm{m}$ diameter liquid water microjets. The pressure waves formed through reflections, at the surface of the microjets, of the initial shock launched in the liquid by the x-ray pulse. These waves developed a characteristic geometric pattern which is related to, but different from, the shock structures of supersonic gas jets. Fully developed waves had initial peak pressures ranging from less than –24 MPa to approximately 100 MPa, which exceed the compressive and tensile strengths of many materials, and correspond to extreme sound intensities on the order of $1\mathrm{GW}/\mathrm{m}{}^2$ and sound pressure levels above 270 dB (re: $1\mu \mathrm{Pa}$). The amplitudes and intensities were limited by the wave destroying its own propagation medium though cavitation, and therefore these ultrasonic waves in jets are one of the most intense propagating sounds that can be generated in liquid water. The pressure of the initial shock decayed exponentially, more rapidly in thinner jets, and the decay length was proportional to the jet diameter within the accuracy of measurements. Extrapolating our results to thinner jets, we find that the pressure waves may damage protein crystals carried by liquid jets in x-ray laser crystallography experiments conducted at megahertz repetition rates.

Reynolds-averaged Navier-Stokes (RANS) models are the primary numerical recourse to investigate complex engineering turbulent flows in industrial applications. However, to establish RANS models as reliable design tools, it is essential to provide estimates for the uncertainty in their predictions. In the recent past, an uncertainty estimation framework relying on eigenvalue and eigenvector perturbations to the modeled Reynolds stress tensor has been widely applied with satisfactory results. However, the methodology for the eigenvector perturbations is not well established. Evaluations using only eigenvalue perturbations do not provide comprehensive estimates of model form uncertainty, especially in flows with streamline curvature, recirculation, or flow separation. In this article, we outline a methodology for the eigenvector perturbations using a predictor-corrector approach, which uses the incipient eigenvalue perturbations along with the Reynolds stress transport equations to determine the eigenvector perturbations. This approach was applied to benchmark cases of complex turbulent flows. The uncertainty intervals estimated using the proposed framework exhibited substantial improvement over eigenvalue-only perturbations and are able to account for a significant proportion of the discrepancy between RANS predictions and high-fidelity data.

The objective of this work is to develop the framework and governing equations needed to analyze the interactions and spectral distribution of internal energy in a manner similar to that of a kinetic energy analysis. To emulate the role of velocity in a kinetic energy analysis, a variable ($\varphi \sim \sqrt{p}$, where $p$ is pressure) is introduced to enable the examination of internal energy dynamics. Evolution equations for the mean and fluctuating components of $\varphi$ are derived. These equations enable the precise examination of mean-turbulent flow internal energy interactions, internal-kinetic energy exchange, and the spectral distribution of internal energy.

At mesoscopic scales electrolyte solutions are modeled by the fluctuating generalized Poisson-Nernst-Planck (PNP) equations [J.-P. Péraud et al., Phys. Rev. Fluids 1, 074103 (2016)]. However, at length and time scales larger than the Debye scales, electrolytes are effectively electroneutral and the charged-fluid PNP equations become too stiff to solve numerically. Here we formulate the isothermal incompressible equations of fluctuating hydrodynamics for reactive multispecies mixtures involving charged species in the electroneutral limit and design a numerical algorithm to solve these equations. Our model does not assume a dilute electrolyte solution but rather treats all species on an equal footing, accounting for cross diffusion and nonideality using Maxwell-Stefan theory. By enforcing local electroneutrality as a constraint, we obtain an elliptic equation for the electric potential that replaces the Poisson equation in the fluctuating PNP equations. We develop a second-order midpoint predictor-corrector algorithm to solve either the charged-fluid or electroneutral equations with only a change of the elliptic solver. We use the electroneutral algorithm to study a gravitational fingering instability, triggered by thermal fluctuations, at an interface where an acid and base react to neutralize each other. Our results demonstrate that, because the four ions diffuse with very different coefficients, one must treat each ion as an individual species and cannot treat the acid, base, and salt as neutral species. This emphasizes the differences between electrodiffusion and classical Fickian diffusion, even at electroneutral scales.

The unsteady behavior of an Edney type-IV shock-dominated hypersonic separated flows over a double-wedge geometry for different gas compositions are studied using the time-accurate direct-simulation Monte Carlo (DSMC) and window proper orthogonal decomposition (WPOD) methods. Near steady state, we find that the POD modes can be correlated with the global modes predicted by linear stability theory. The WPOD analyses show that the first mode for each flow quantity outlines the corresponding steady-state solution and that higher spatial POD modes are pronounced in the bow shock, separation, and transmitted shocks and shear layers, thereby coupling these regions. The temporal modes corresponding to each macroscopic flow quantities are observed to have the same decay rates. The effects of three types of gas compositions—molecular nitrogen, nonreacting air consisting of molecular nitrogen and oxygen, and reacting air with oxygen dissociation and the ${\mathrm{N}}_2$ + O exchange reactions—on flow stability are considered. Nonequilibrium thermochemical effects are found to change the shock structures, the size of the separation region, and the time required to reach steady state. The decay rate of the least damped eigenmode for the chemically reacting air case is found to be smaller in comparison to the nonreacting air case since the translational temperatures downstream of the bow shock are lower due to endothermic chemical reactions. The simulated heat fluxes and shock standoff distance are found to be in qualitative agreement with recent experiments where the freestream density is eight times higher than the current DSMC computations.

A bi-frequency open-loop control strategy aiming to combine both high- and low-frequency forcing effects is used to experimentally reduce the drag of a simplified car model at a slight yaw angle of $5^{\circ }$. The unforced mean wake features a lateral asymmetry which induces a low base pressure footprint close to the leeward side and increases drag compared to the aligned model. Forcing is performed with pulsed jets along the windward trailing edge. High-frequency forcing acts as a time-invariant flap. The fluidic flap effect deviates the windward shear layer towards the leeward side and reduces the wake bluffness, but the lateral asymmetry of the near wake is still observed. The drag reduction related to this high-frequency forcing is about 6% with a high actuation efficiency. A modulation of the high-frequency forcing with a low-frequency component is then introduced in order to modify the mass and momentum exchange in the separating shear layer at the windward trailing edge. We find that the modulated forcing provides the ability to manipulate the mean wake orientation while maintaining the fluidic flap effect. Among all wake orientations, those reducing drag are the ones having a mean symmetric wake. The bi-frequency control strategy leads to a maximum drag reduction of 7% for the best choice of frequencies. Importantly, the bi-frequency control is more efficient than the single high-frequency forcing, the actuator requiring only half the actuation energy and presenting an actuation efficiency multiplied by 3. Finally, the physical mechanisms related to drag reduction are carefully analyzed. In particular, we show that the wake symmetrization reduces the global production of turbulent kinetic energy in the shear layers. These results open up opportunities for closed-loop control of wake asymmetries.

The aerodynamic loads on a static airfoil are modified below its stall margin by deliberate formation and regulation of trapped vorticity on the suction and pressure surfaces near the trailing edge. Vorticity accumulation and shedding are effected using a hybrid actuator comprising a miniature two-dimensional passive protuberance having a cross-stream scale that is nominally commensurate with the local boundary layer thickness, and the vorticity flux is regulated using integrated spanwise arrays of equally spaced high aspect ratio synthetic jet actuators. The aerodynamic loads are varied by independent differential control of flux of opposite sense vorticity from both sides of the airfoil that are characterized using measurements of the force/torque, surface pressure, and particle image velocimetry over the suction surface and in the airfoil's near wake. The induced changes in circulation and lift and in the pitching moment are traced to complex transitory changes in vorticity flux associated with the onset and termination of the actuation. These changes are characterized by transitory variations in the balance of flux and shedding of clockwise and counterclockwise vorticities that are triggered, respectively, by actuation on the pressure or suction surfaces. These investigations indicate that hybrid flow control by trapped vorticity, which has the potential to manipulate prestall aerodynamic loads without mechanical control surfaces, may enable novel wing designs.

When transported in confined geometries rigid fibers show interesting transport dynamics induced by friction with the top and bottom walls. Fiber flexibility causes an additional coupling between fiber deformation and transport and is expected to lead to more complex dynamics. A first crucial step for their understanding is the characterization of the deformed fiber shape. Here we characterize this shape for a fiber transported in a confined plug flow perpendicular to the flow direction using a combination of microfluidic experiments and numerical simulations. In the experiments, size, initial orientation, and mechanical properties of the fibers are controlled using microfabrication techniques and in situ characterization methods. The numerical simulations use modified Brinkman equations as well as full three-dimensional simulations. We show that the bending of a perpendicular fiber results from the force distribution acting on the elongated object and is proportional to the elasto-viscous number, which compares viscous to elastic forces. We quantitatively characterize the influence of the confinement on the fiber deformation. The precise understanding of the deformation of a flexible fiber in a confined geometry can also be used in future to understand the deformation and transport of more complex deformable particles in confined flows, such as vesicles or red blood cells.

Experiments were conducted with three-dimensional printed disks and rods to study the orientation dynamics of nonspherical particles under surface gravity waves. Trials were run with both neutrally buoyant and slightly negatively buoyant particles, which were large enough that inertia due to their finite size was important. Although the particles had a broad distribution of initial orientations, over time the waves were observed to focus these orientations toward a preferred angle that agreed with theory. Negatively buoyant particles additionally exhibited a tendency to adopt an orientation that maximized vertical drag. The overall orientation can be described as the result of a competition between the orientation favored by waves and the orientation favored by settling. The spread about the mean orientation was also observed to increase with wave strength. Finally, the stabilization of out-of-plane orientations of disk-shaped particles was observed due to their finite size.

We present an experimental analysis of the linear and nonlinear regimes of an attractor of inertial waves in a trapezoidal cavity under rotation. Varying the rotation rate and the forcing amplitude and wavelength, we identify the scaling laws followed by the attractor amplitude and wavelength in both regimes. In particular, we show that the nonlinear scaling laws can be well described by replacing the fluid viscosity in the linear model by a turbulent viscosity, a result that could help in extrapolating attractor theory to geophysically and astrophysically relevant situations. We further study the triadic resonance instability of the attractor which is at the origin of the turbulent viscosity. We show that the typical frequencies of the subharmonic waves produced by the instability behave very differently from previously reported numerical results and from the prediction of the theory of triadic resonance. This behavior might be related to the deviation from the horizontal invariance of the attractor in our experiment in relation to the presence of vertical walls of the cavity, an effect that should be at play in all practical situations.

To date, axisymmetric internal wave fields, which have relevance to atmospheric internal wave fields generated by storm cells and oceanic near-inertial wave fields produced by surface perturbations, have been experimentally realized using an oscillating sphere or torus as the source. Here we use a wave generator configuration capable of exciting axisymmetric internal wave fields of arbitrary radial form to generate axisymmetric internal wave modes. After establishing the theoretical background for axisymmetric mode propagation, taking into account lateral and vertical confinement, and also accounting for the effects of weak viscosity, we study modes of different order. We characterize the efficiency of the wave generator through careful measurement of the wave amplitude based upon group velocity arguments, and then consider the effect of vertical confinement to induce resonance, identifying a series of experimental resonant peaks that agree well with theoretical predictions. In the vicinity of resonance, the wave fields undergo a transition to nonlinear behavior that is initiated on the central axis of the domain and proceeds to erode the wave field throughout the domain.

Sorting cells based on their intrinsic properties is a highly desirable objective, since changes in cell deformability are often associated with various stress conditions and diseases. Deterministic lateral displacement (DLD) devices offer high precision for rigid spherical particles, while their success in sorting deformable particles remains limited due to the complexity of cell traversal in DLD devices. We employ mesoscopic hydrodynamics simulations and demonstrate prominent advantages of sharp-edged DLD obstacles for probing deformability properties of red blood cells (RBCs). By consecutive sharpening of the pillar shape from circular to diamond to triangular cross-section, a pronounced cell bending around an edge is achieved, serving as a deformability sensor. Bending around the edge is the primary mechanism which governs the traversal of RBCs through such a DLD device. This strategy requires an appropriate degree of cell bending by fluid stresses, which can be controlled by the flow rate, and exhibits good sensitivity to moderate changes in cell deformability. We expect that similar mechanisms should be applicable for the development of novel DLD devices that target intrinsic properties of many other cells.

Particle image velocimetry and particle tracking velocimetry have been employed to investigate the dynamics of finite-size spherical particles, slightly heavier than the carrier fluid, in a horizontal turbulent square duct flow. Interface resolved direct numerical simulations (DNSs) have also been performed with the immersed boundary method at the same experimental conditions, bulk Reynolds number ${\text{Re}}_{2H}=5600$, duct height to particle-size ratio $2H/d_p=14.5$, particle volume fraction $\mathrm{\Phi }=1\%$, and particle to fluid density ratio ${\rho }_p/{\rho }_f=1.0035$. Good agreement has been observed between experiments and simulations in terms of the overall pressure drop, concentration distribution, and turbulent statistics of the two phases. Additional experimental results considering two particle sizes $2H/d_p=14.5$ and 9 and multiple $\mathrm{\Phi }=1\%$, 2%, 3%, 4%, and 5% are reported at the same ${\text{Re}}_{2H}$. The pressure drop monotonically increases with the volume fraction, almost linearly and nearly independently of the particle size for the above parameters. However, despite the similar pressure drop, the microscopic picture in terms of fluid velocity statistics differs significantly with the particle size. This one-to-one comparison between simulations and experiments extends the validity of interface resolved DNS in complex turbulent multiphase flows and highlights the ability of experiments to investigate such flows in considerable detail, even in regions where the local volume fraction is relatively high.